Uniform hyperbolicity of invariant cylinder

Cheng, Chuan

doi:10.4310/jdg/1493172093

Cited by 13 publications

(15 citation statements)

References 7 publications

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“…In [37] an autonomous case n ≥ 4 is considered and methods that can be used in this situation are discussed. In [14] a proof for n ≥ 4 is announced (see also [15,16,17]).…”

Section: Introductionmentioning

confidence: 99%

Arnold diffusion in multidimensional a priori unstable Hamiltonian systems

Davletshin,

Treschev

2018

Preprint

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We study the Arnold diffusion in a priori unstable near-integrable systems in a neighbourhood of a resonance of low order. We consider a nonautonomous near-integrable Hamiltonian system with n + 1/2 degrees of freedom, n ≥ 2. Let the Hamilton function H of depend on the parameter ε, for ε = 0 the system is integrable and has a homoclinic asymptotic manifold Γ. Our main result is that for small generic perturbation in an ε-neighborhood of Γ there exist trajectories the projections of which on the space of actions cross the resonance. By "generic perturbations" we mean an open dense set in the space of C r -smooth functions d dε ε=0 H, r = r 0 , r 0 + 1, . . . , ∞, ω. Combination of this result with results of [44] answers the main questions on the Arnold diffusion in a priori unstable case: the diffusion takes place for generic perturbation, diffusion trajectories can go along any smooth curve in the action space with average velocity of order ε/| log ε|.

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“…In [37] an autonomous case n ≥ 4 is considered and methods that can be used in this situation are discussed. In [14] a proof for n ≥ 4 is announced (see also [15,16,17]).…”

Section: Introductionmentioning

confidence: 99%

Arnold diffusion in multidimensional a priori unstable Hamiltonian systems

Davletshin,

Treschev

2018

Preprint

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“…这是一个不可积系统. 利用变分方法我们证明 (参见文献 [82]) [81]). 注意到 Mañé 集Ñ (C) 位于 E = α(c) 的能量面上, 我们建立了这样一种上同调等价关系: [28,83]).…”

Section: Kolmogorov 定理unclassified

Dynamical diversity of nearly integrable Hamiltonian systems

Chong-Qing¹

2020

Sci. Sin.-Math.

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The dynamics of nearly integrable Hamiltonian is called by Poincaré the fundamental problem of dynamical systems. Since the fifties of the last century, the great achievements have been made in the study of the problem. Among them, the Kolmogorov's theorem on invariant tori and the discovery of Arnold diffusion are the landmarks. They have greatly deepened our understanding on the dynamical diversity of nearly integrable Hamiltonian systems. We shall briefly review some of the developments.

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“…So, it is a challenge to construct cylinder in such a disc. The condition n = 3 allows us to apply a variational method to construct cylinder which extends o( √ )-close to double resonant point, see [CZ,C17a]. Because of the result and by a new cohomology equivalence, we found a way in [C17b] to pass through the small neighborhood by turning around the strong double resonant point and joining two cylinders.…”

Section: Introductionmentioning

confidence: 97%

“…The weak double resonance can be reduced to a priori unstable case such the problem is reduced to the finite number of strong double resonances. In Section 6, we construct transition chain crossing strong double resonances by applying the main results of [CZ,C17a,C17b]. By preparing some technical estimates for the nearly integrable system including the deviation of the rotation vectors, the location of the flat and the estimate of orbits in the Aubry sets in Section 7, we prove Theorem 2.1 in Section 8.…”

Section: Introductionmentioning

confidence: 99%

The genericity of Arnold diffusion in nearly integrable Hamiltonian systems

Cheng¹

2019

Asian Journal of Mathematics

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In this paper, we prove that the net of transition chain is δ-dense for nearly integrable positive definite Hamiltonian systems with 3 degrees of freedom in the cusp-residual generic sense in C r -topology, r ≥ 6. The main ingredients of the proof existed in [CZ, C17a, C17b]. As an immediate consequence, Arnold diffusion exists among this class of Hamiltonian systems. The question of [C17c] is answered in Section 9 of the paper.C a = {λP : P ∈ R a , λ ∈ R P }.

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Uniform hyperbolicity of invariant cylinder

Abstract: Abstract. For a nearly integrable Hamiltonian systems H = h(p) + ǫP (p, q) with

Cited by 13 publications

References 7 publications

Arnold diffusion in multidimensional a priori unstable Hamiltonian systems

Arnold diffusion in multidimensional a priori unstable Hamiltonian systems

Dynamical diversity of nearly integrable Hamiltonian systems

The genericity of Arnold diffusion in nearly integrable Hamiltonian systems

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